The dataset and the implementation used for the above paper can be found here and here respectively. Facts are:

the approach and the implementation is not sound and complete, view this example and the explanation provided within, together with files test.csp and test.mgraph.

the evaluation makes improper use of competing solvers and does not improve the state-of-the-art, view this paper. (At
the time of writting the paper we were not aware of the theoretical/algorithmical flaws, their current implementation would objectively perform worse had the partitioning been proper and all necessary constraint checks been performed. So, what we describe
in Section II.C in our paper is what the partitioning should be like, and not how it is in the AAAI paper. A more thorough explanation for a proper partitioning is provided in this paper. In short, the patchwork property that is used in both graph decomposition works relies on tree
decompositions to guarantee soundness and completeness. Intuitively, tree decompositions are used as a requirement of acyclicity of cliques. The AAAI paper does not use tree decompositions.)

The authors have partially acknowledged the soundness and completeness issue with a note on their website: aaai2014-note.txt.